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LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       465
until the remaining ones ate zero.For example, for a fourth-degree polynornialin s,
as - a6 -.  .- .  - 0 so thatb3 - b4 =  ...  - O,c2 - C3 =  ...  = O,d2 - d3 - ...  = O,
e2-e3-. "- =O,and f2- f3= .*. -.0.
    Routh's stability criterion states that the number of roots of the characteristic
polynomial with positive real parts is equal to the number of changes in sign of
the coefficients of the first column of Routh's array.lt is important to note that the
exact values of these coefficients need not be known; instead only the signs are
required. Thus, the sufficiency condition for a closed-loop system to be stable is
that all the elements of the first column of Routh's array must be positive or must
have the same sign.
      To summarize, the necessary and sufficient condition for the stability ofa closed-
loop system is that all the coeffcients of the characteristic polynomial and the
elements of the frrst column of Routh's array must be positive or must have the
same sign.
    If any of the coefficients of the characteristic polynomial or any element 'of
the first column in the Routh's array is zero, then replace that term by a verj(
small positive number 6 and proceed as usual with the evaluation of the rest of the
elements of Routh's array.
   For a fourth-order polynomial, the Routh's stability criterion reduces .to the
following:
     1) All coefficients ao, ai, a2, a3, and a4 musl         lositive.
    2) The Routh's discriminant (ala2 - aOa3)a~ b  p.aS4 must be positive.
Example 5.2
      Using Routh's criterion, determine the stability of the system represented by the
following characteristic polynomial:
s4+2s3+5s2+2s+2 = O
       Solution.      Thisis afourth-degree polynomialin s,We have ao - 1, ai - 2, a2 -
5, a3 = 2, a4 = 2, and as -.O. Because all the coefficients are positive and none of
the coefficients ao to a4 is zero, the necessary condition for stability is satisfied.
To examine whether the sufficiency condition is satisfied, we form Routh's array
as follows:
S4: 1 5 2
S3: 2 2 O
S2: 4 2 0
Sl: 1 O
so: 2
We observe that all the elements of the first column of this table are positive; hence
the sufficiency condition is also satisfied. Hence, the characteristic polynomial has
no positive real root or a complex root with positive real part and the given system
is stable.
 .
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it
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--
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i,
Sn-I ai a3 as a7
Sn-2 bi b2 b3 b4
Srt-3 Cl C2 C3 C4
Sn-4 di d2 d3 d4
ii
...
S2  ei e2
Sl  f]
SO gl
466           PERFORMANCE, STABIL17Y, DYNAMICS, AND CONTROL
Example 5.3
Given the characteristic polynomial
S4 +3S3 +2S2 +4s +1 -. O
Examine the stability of the system using Routh's stability criterion.
   So/ution.  Because all of the coefficients of this fourth-degree polynomial
are positive, the necessary condition is satisfied. To see whether the sufficiency
condition is satisfied, form Routh's array as follows:
S4
S3
S2 :
Sl
SO :
There are two sign changes in the first column starting with the row corresponding
to S2. Hence, there will be two roots that are either positive or have positive real
parts and the given system is unstable.
                                             Example 5.4
   For the system whose characteristic polynomialis given by
                                  s4+2s2+5s+2 = 0
Examine the stability of the system using Routh's criterion.
   So/ution.  Notice that the S3 term is missing. Hence, we rewrite the given
polynomial as follows:
s4+€s3+2s2+5s+2 = 0
where e is a small positive number, say 0.0001. With this, we observe that the nec-
essary condition is satisfied. To see whether the sufficiency condition is satisfied,
we form Routh's array as follows:
　
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